Finite Group Databasebeta

← #16923489

Subgroups — #16923489

Order 4352.

366 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14530327C211yesmaximal, characteristic
2#14530260C211yesmaximal, characteristic
3#14530255C211yesmaximal, characteristic
4#14526610C211yesmaximal, characteristic
5#14526607C211yesmaximal, characteristic
6#14526607C211yesmaximal, characteristic
7C17×#2805C211yesmaximal, characteristic
8C2×#10587478—21no—
9C2×#10587479—21no—
10C2×#10587479—21no—
11C2×#10587506—21no—
12#11762679C2211yescharacteristic
13#11762571—21no—
14#11762571—21no—
15#11762564C2211yescharacteristic
16#11762564C2211yescharacteristic
17#11762564C2211yescharacteristic
18C17×#400—21no—
19C17×#400—21no—
20C17×#395—21no—
21C17×#391C2211yescharacteristic
22C17×#389—21no—
23C17×#389C2211yescharacteristic
24C17×#389C2211yescharacteristic
25C17×#387—21no—
26C17×#387—21no—
27C22×Q8×C17D411yescharacteristic
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C42×C17—21no—
38C2×C42×C17C2311yescharacteristic
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C22×#3622—21no—
43C22×#3622—21no—
44C22×#3622—21no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95—21no—
54#10587506—42no—
55#10587479—42no—
56#10587479—42no—
57#10587478—42no—
58C2×Q8×C17—21no—
59C2×Q8×C17—21no—
60C2×Q8×C17—41no—
61C2×Q8×C17—41no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—41no—
72C4⋊C4×C17—21no—
73C4⋊C4×C17—21no—
74C4⋊C4×C17—21no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—81no—
83C4⋊C4×C17—81no—
84C4⋊C4×C17SD1612yes—
85C42×C17—41no—
86C42×C17—81no—
87C4⋊Dic17—41no—
88C4⋊Dic17—41no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93C2×#3622—22no—
94C2×#3622—22no—
95C2×#3622—22no—
96C2×#3622—41no—
97C2×#3622—41no—
98C2×#3622—41no—
99C2×#3622—41no—
100C2×#3622—41no—
101C2×#3622—41no—
102#7807—171nomaximal, Sylow
103C23×C17#12011yescharacteristic
104Q8×C17—42no—
105Q8×C17—81no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—22no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17C2×SD1611yescharacteristic
131C2×C4×C17#13511yescharacteristic
132C2×C4×C17#13511yescharacteristic
133C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C17#13711yescharacteristic
135C2×C4×C17#13711yescharacteristic
136C2×Dic17—81no—
137C2×Dic17—81no—
138C2×Dic17—81no—
139#3622—42no—
140#3622—42no—
141#3622—42no—
142#2805D1711yescharacteristic
143#2096—171no—
144#2084—171no—
145#2083—171no—
146#1272—171no—
147#1271—171no—
148#1271—171no—
149C22×C17#44811yescharacteristic
150C22×C17#44911yescharacteristic
151C22×C17#45211yescharacteristic
152C22×C17#45311yescharacteristic
153C22×C17#45611yescharacteristic
154C22×C17#45611yescharacteristic
155C22×C17#45811yescharacteristic
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—42no—
163C4×C17—81no—
164C4×C17—81no—
165C4×C17—81no—
166C4×C17—81no—
167C4×C17—81no—
168Dic17—161no—
169C2×#107—341no—
170C2×#106—341no—
171C2×#106—341no—
172C2×#103—341no—
173#400—21no—
174#400—21no—
175#395—21no—
176#391C2×D1711yescharacteristic
177#389—21no—
178#389C2×D1711yescharacteristic
179#389C2×D1711yescharacteristic
180#387—21no—
181#387—21no—
182#384—171no—
183#340—341no—
184#340—341no—
185#336—171no—
186#336—171no—
187#336—171no—
188C2×C17#160911yescharacteristic
189C2×C17#161111yescharacteristic
190C2×C17#161811yescharacteristic
191C2×C17#162211yescharacteristic
192C2×C17#165011yescharacteristic
193C2×C17#165311yescharacteristic
194C2×C17#166011yescharacteristic
195C22×Q8C22⋊D1711yescharacteristic
196C22×C8—341no—
197C22×C8—341no—
198C22×C8—341no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4C22⋊D1711yescharacteristic
207C2×C4⋊C4C22⋊D1711yescharacteristic
208C2×C4⋊C4C22⋊D1711yescharacteristic
209C2×C4⋊C4C22⋊D1711yescharacteristic
210C2×C4⋊C4C22⋊D1711yescharacteristic
211C2×C42—21no—
212C2×C42C22×D1711yescharacteristic
213#107—682no—
214#106—682no—
215#106—682no—
216#103—682no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226C17#780711yescharacteristic, complemented, Sylow
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—41no—
230C2×Q8—41no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—681no—
238C22×C4C2×C22⋊D1711yescharacteristic
239C22×C4C2×C22⋊D1711yescharacteristic
240C22×C4C2×C22⋊D1711yescharacteristic
241C2×C8—342no—
242C2×C8—342no—
243C2×C8—342no—
244C2×C8—681no—
245C2×C8—681no—
246C2×C8—681no—
247C2×C8—681no—
248C2×C8—681no—
249C2×C8—681no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—681no—
263C4⋊C4—681no—
264C4⋊C4—681no—
265C4⋊C4—681no—
266C4⋊C4—681no—
267C4⋊C4—681no—
268C4⋊C4#6323012yes—
269C42—41no—
270C42—81no—
271C23#1058763911yescharacteristic
272Q8—42no—
273Q8—81no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—1361no—
299C2×C4—1361no—
300C2×C4—1361no—
301C2×C4C2×#6323011yescharacteristic
302C2×C4#1058762811yescharacteristic
303C2×C4#1058763511yescharacteristic
304C2×C4#1058763611yescharacteristic
305C2×C4#1058763611yescharacteristic
306C2×C4#1058763711yescharacteristic
307C8—682no—
308C8—682no—
309C8—682no—
310C22#1176294811yescharacteristic
311C22#1176295411yescharacteristic
312C22#1176297111yescharacteristic
313C22#1176297711yescharacteristic
314C22#1176299211yescharacteristic
315C22#1176299211yescharacteristic
316C22#1176299911yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—2721no—
330C2#1452778111yescharacteristic
331C2#1452779211yescharacteristic
332C2#1452783211yescharacteristic
333C2#1452785611yescharacteristic
334C2#1452804011yescharacteristic
335C2#1452806111yescharacteristic
336C2#1452809511yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2805C2
socle, prime radicalC23×C17#120
hypercenter#2805D17
hypocenterC17#7807
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2805

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
226. C17#7807semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145303272
#145302602
#145302552
#145266102
#145266072
C17×#28052
#780717